Eleanor
I haven't used the self-adhesive foam you asked about but I can provide some general info about adhesives and how to use them. I am a sign maker by trade and use adhesive vinyl films, from which I cut the graphics and lettering that I apply to sign substrates. So applying a print to foam board is actually the same process I use every day when making signs.
Having a clean, dust free environment is extremely important.
Before actually doing the process "for real" go through it completely step-by-step as though you were actually doing it but without removing the protective paper from the adhesive. You want to make sure you decide how you are going to line the print up on top of the piece of board. Are you going to apply the print to a larger foam board and then cut off any excess of board or put it on a piece the same size? Putting the print on the larger board is easier since it is not as critical when lining the print to the board. You don't get a second chance (unless you are using a "re-positional adhesive) to realign the print as once the print touches the adhesive it is near impossible to remove it without leaving some torn paper from the back of the print attached to the adhesive. So you get one opportunity to align your print properly. Let the print lay flat on the foam board and do not apply finger pressure to push the print flat.
I typically use a straight edge positioned on the foam board to which I align the print. This allows me to be very precise in the placement of the print on the foam board. If you're using a piece of foam board that is larger than the print take a piece of the protective adhesive paper (silicone side towards the adhesive) and use that between the straight edge and the surface of the adhesive to prevent the straight edge from adhering to your foam board.
When you peel the protective paper off the foam board you create a static charge that attracts dust and dirt like crazy to the freshly exposed adhesive surface. Be careful not to create a "wind" when moving the foam board and print as this will stir up dust which inevitably lands on the adhesive.
Did I mention that it was extremely important to have a clean, dust free environment?
In order for anything to remain "stuck permanently" you need to make sure that the print and the adhesive side of the foam board are pressed firmly together. You must remember that foam board is compressible and that “too much pressure” cannot be used. However insufficient pressure will cause the print to pop off in areas or come loose completely from the foam board over time. You should take some scrap foam board and practice applying pressure to it to see how much pressure you can apply before damaging i.e. compressing the foam board.
I will typically use one of two methods for adhering the print to the foam board. Predominantly I use a squeegee, see one here
http://www.speedpress.com/products/1/54 . The way this is used, is once the print is placed on top of the foam board I put a protective piece of paper over the print and then working from the center of the print out toward the edge I apply pressure with the squeegee making sure to overlap each successive stroke of the squeegee to ensure that the pressure is applied uniformly over the entire print. Going back and repeating the process a second or third time will ensure that you have adhered everything permanently. Stroking the squeegee firmly across the entire print insures that your print is securely stuck to the adhesive. It also ensures that any “trapped air” is squeegee out to the edge of the print. Uneven pressure and non-overlapping strokes of the squeegee can allow air to be trapped between the back of the print and the surface of the foam board. Writing about this procedure makes it seem complicated but in reality it is very easy! Once you have practice it a couple of times it will be a very easy to do.
However I would advise anyone doing this for the first time not to start out with a big non-replaceable print but rather use a small print that is easily replaced, should something unforeseen happen.
The second method I use is very similar with the exception that a roller is used instead of the squeegee. The roller is typically 3 to 4 inches wide and an 1” to 1.5” in diameter. I prefer a roller that has some give to it, as opposed to one that is very hard uncompressible rubber. The roller is used in the same manner as this squeegee i.e. you work from the center towards the edge with overlapping strokes.